Bifurcation and stability analyses for a coupled Brusselator model

P. Yu, Abba Gumel

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

This paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two Brusselators. Of particular interest is the study of the associated Hopf bifurcation and double-Hopf bifurcations. The motion of the oscillator may either be periodic (bifurcating from a Hopf-type critical point), or quasi-periodic (bifurcating from a compound critical point). Furthermore, bifurcation analysis reveals that the limit cycles associated with the first Brusselator are always stable, while that generated by the second Brusselator may be unstable if the parameter values are chosen far from the stability boundary. It is interesting to note that in the vicinity of the double-Hopf compound critical point, there exist periodic as well as quasi-periodic solutions. The quasi-periodic motion is stable for a small parameter region. A robust Gauss-Seidel like implicit finite-difference method (GS1) has been developed and used for the solution of the resulting initial-value problem (IVP). In addition to being of comparable accuracy (judging by the similarity of the profiles generated) with the fourth order Runge-Kutta method (RK4), the GS1 method will be seen to have better numerical stability property than RK4. Unlike the RK4, which fails when large time steps are used to integrate the IVP, extensive numerical simulations with appropriate initial data suggest that the GS1 method is unconditionally convergent. Moreover, it is more economical computationally.

Original languageEnglish (US)
Pages (from-to)795-820
Number of pages26
JournalJournal of Sound and Vibration
Volume244
Issue number5
DOIs
StatePublished - Jul 26 2001
Externally publishedYes

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Hopf bifurcation
Initial value problems
critical point
boundary value problems
Runge Kutta methods
Convergence of numerical methods
oscillators
Finite difference method
numerical stability
Runge-Kutta method
Computer simulation
cycles
profiles
simulation

ASJC Scopus subject areas

  • Engineering(all)
  • Mechanical Engineering

Cite this

Bifurcation and stability analyses for a coupled Brusselator model. / Yu, P.; Gumel, Abba.

In: Journal of Sound and Vibration, Vol. 244, No. 5, 26.07.2001, p. 795-820.

Research output: Contribution to journalArticle

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